A Floer-theoretic interpretation of the polynomial representation of the double affine Hecke algebra

Abstract

We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of $κ$-tuples of cotangent fibers and $κ$-tuples of conormal bundles of homotopically nontrivial simple closed curves in $T^*Σ$ with a certain braid skein group, where $Σ$ is a closed oriented surface of genus $> 0$ and $κ$ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to $Σ$. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where $Σ=T^2$ and the cotangent fibers and conormal bundles of curves are both parallel copies.